Asymptotic behavior of solutions for the non-autonomous classical reaction-diffusion equation with nonlinear boundary conditions and fading memory

LIANG Yuting; WANG Xuan

doi:10.3969/j.issn.1000-5641.201911046

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LIANG Yuting, WANG Xuan. Asymptotic behavior of solutions for the non-autonomous classical reaction-diffusion equation with nonlinear boundary conditions and fading memory[J]. Journal of East China Normal University (Natural Sciences), 2021, (1): 16-27. doi: 10.3969/j.issn.1000-5641.201911046

Citation:

LIANG Yuting, WANG Xuan. Asymptotic behavior of solutions for the non-autonomous classical reaction-diffusion equation with nonlinear boundary conditions and fading memory[J]. Journal of East China Normal University (Natural Sciences), 2021, (1): 16-27. doi: 10.3969/j.issn.1000-5641.201911046

Citation:

LIANG Yuting, WANG Xuan. Asymptotic behavior of solutions for the non-autonomous classical reaction-diffusion equation with nonlinear boundary conditions and fading memory[J]. Journal of East China Normal University (Natural Sciences), 2021, (1): 16-27. doi: 10.3969/j.issn.1000-5641.201911046

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Asymptotic behavior of solutions for the non-autonomous classical reaction-diffusion equation with nonlinear boundary conditions and fading memory

doi: 10.3969/j.issn.1000-5641.201911046

LIANG Yuting,
WANG Xuan^,

College of Mathematics and Statistics, Northwest Normal University, Lanzhou　730070, China

Received Date: 2019-11-25
Publish Date: 2021-01-27

Abstract

Abstract

In this paper, we study the long-time dynamic behavior of solutions for the non-autonomous classical reaction-diffusion equation with nonlinear boundary conditions and fading memory, where the internal nonlinearity and boundary nonlinearity adheres to polynomial growth of arbitrary order as well as the balance condition. In addition, the forcing term is translation bounded, rather than translation compact, by use of contractive function method and process theory. The existence and the topological structure of uniform attractors in $L^{2}(\Omega)\times L_\mu^2(\mathbb R^+; H_{0}^1(\Omega))$ are proven. This result extends and improves existing research in the literature.
- non-autonomous classical reaction-diffusion equation,
- uniform attractors,
- nonlinear boundary,
- fading memory,
- polynomial growth of arbitrary order

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References(18)

References

[1]	DAFERMOS C M. Asymptotic stability in viscoelasticity [J]. Archive Rational Mechanics Analysis, 1970, 37(4): 297-308. DOI: 10.1007/BF00251609.
[2]	YANG L. Uniform attractors for the closed process and application to the reaction-diffusion equation with dynamical boundary condition [J]. Nonlinear Anal, 2009, 71: 4012-4025. DOI: 10.1016/j.na.2009.02.083.
[3]	YANG L. Asymptotic regularity and attractors of the reaction-diffusion equation with nonlinear boundary condition [J]. Nonlinear Analysis: Real World Applications, 2012, 13(3): 1069-1079. DOI: 10.1016/j.nonrwa.2011.02.024.
[4]	YANG L, YANG M H. Attractors of the non-autonomous reaction-diffusion equation with nonlinear boundary condition [J]. Nonlinear Analysis: Real World Applications, 2010, 11(5): 3946-3954. DOI: 10.1016/j.nonrwa.2010.03.002.
[5]	ZHONG C K, YANG M H, SUN C Y. The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations [J]. Journal of Differential Equations, 2006, 223: 367-399. DOI: 10.1016/j.jde.2005.06.008.
[6]	SUN C Y, WANG S H, ZHONG C K. Global attractors for a nonclassical diffusion equation [J]. Acta Math Sin(Engl Ser), 2007, 23: 1271-1280. DOI: 10.1007/s10114-005-0909-6.
[7]	SUN C Y, YANG M H. Dynamics of the nonclassical diffusion equation [J]. Asymptot Anal, 2008, 59: 51-81. DOI: 10.3233/ASY-2008-0886.
[8]	SONG H T, MA S, ZHONG C K. Attractors of non-autonomous reaction-diffusion equations [J]. Nonlinearity, 2009, 22: 667-681. DOI: 10.1088/0951-7715/22/3/008.
[9]	CHEPYZHOV V V, MIRANVILLE A. On trajectory and global attractors for semilinear heat equations with fading memory [J]. Indiana University Mathematics Journal, 2006, 55(1): 119-168. DOI: 10.1512/iumj.2006.55.2597.
[10]	GIORGI C, PATA V. Asymptotic behavior of a nonlinear hyperbolic heat equation with memory [J]. Nonlinear Differential Equations Applications Nodea, 2001, 8(2): 157-171. DOI: 10.1007/PL00001443.
[11]	汪璇, 朱宗伟, 马巧珍. 带衰退记忆的经典反应扩散方程的全局吸引子 [J]. 数学年刊(中文版), 2014, 35(4): 423-434.
[12]	PATA V, SQUASSINA M. On the strongly damped wave equation [J]. Communications in Mathematical Physics, 2005, 253(3): 511-533. DOI: 10.1007/s00220-004-1233-1.
[13]	PATA V, ZUCCHI A. Attractors for a damped hyperbolic equation with linear memory [J]. Advances in Mathematical Sciences and Applications, 2001, 11(2): 505-529.
[14]	RODRÍGUEZ-BERNAL A, TAJDINE A. Nonlinear balance for reaction diffusion equations under nonlinear boundary conditions: Dissipativity and blow-up [J]. Journal of Differential Equations, 2001, 169(2): 332-372. DOI: 10.1006/jdeq.2000.3903.
[15]	SUN C Y, CAO D M, DUAN J Q. Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity [J]. Nonlinearity, 2006, 19(11): 2645-2665. DOI: 10.1088/0951-7715/19/11/008.
[16]	XIE Y Q, LI Y A, ZENG Y. Uniform attractors for nonclassical diffusion equations with memory[J]. Journal of Function Spaces, 2016 (3/4): 1-11.
[17]	CHEPYZHOV V V, VISHIK M I. Attractors for Equations of Mathematical Physics [M]. Providence, RI: Amer Math Soc, 2001.
[18]	SUN C, CAO D, DUAN J. Uniform attractors for non-autonomous wave equations with nonlinear damping [J]. SIAM Journal on Applied Dynamical Systems, 2007, 6(2): 293-318. DOI: 10.1137/060663805.